Calculus Review Derivatives Objectives At the end of this lesson you should be able to 1 Find the derivative of functions involving sums products quotients and composition of polynomial exponential and logarithmic structures 2 Evaluate the derivative find the slope at any point in the domain of the function Background You need to review the formulas commonly used in finding derivatives I know you don t like the technical definition It took me about a year to get it down really solid However there are times in business where you must use the definition because the problem set is discrete Here s the formula again f x lim h 0 f x h f x h Just as a reminder it says 1 Calculate the functional value at a point in domain x and at a point h units distant from it This creates the change in output or rise 2 Create the ratio of the result of step 1 and h Now you have rise over run or the average rate of change across the h units of domain from x to x h 3 If it is possible shrink h to as close to zero as possible That creates the functions limit at point x 4 If all the math works we have the derivative also known as the instantaneous rate of change the slope of the tangent line or the rate of change at the point x and maybe another three or four common names Let s get blunt If this is a mystery to you you need to get help now From this powerful calculation we developed a host of the following rules The Power Rule d n x n x n 1 dx Where u f x v g x are both continuous functions we get these results 1 Product Rule d u v u v u v dx 2 Quotient Rule d u u v u v dx v v2 3 Chain Rule dy dy du dx du dx and the Generalized Power Rule d n u n u n 1 u dx Arizona State University Department of Mathematics and Statistics 1 Calculus Review Derivatives We created special cases to simplify our work They all come out of the Chain Rule which may explain why I ve never really memorized them 1 Exponential Rules d x d u d x e e x and e eu u a a x ln a dx dx dx d u a a u ln a u dx 2 Logarithmic Rules d 1 d u ln x and ln u dx x dx u Finally if y f x the derivative can be written as y f x dy df x dx dx Discussion We developed the process of finding the derivatives of non linear functions because the slope of a curve changes at each point on the graph As we said the formula for finding the first derivative is f x lim h 0 was developed from the original formula m f x h f x Remember how it h y2 y1 for finding the slope of a line joining two points x2 x1 x1 y1 and x2 y2 All other formulas we developed later were based on this The only problem with the formula is that it has too many names The derivative is defined to be the slope of the tangent line at any point on the graph so long as it is a unique value It is also considered as the rate of change of the variable y with respect to the variable x Recall that the simple function f x x is beautifully continuous However it does not have a derivative at zero because the slope of the tangent line has no unique value there Example Suppose we have the function f x 2 x 2 5 x 4 From algebra you know this is a convex quadratic function that is that opens up The slope is negative on the left of the vertex and positive on the right of the vertex However the slope is different at every point on the graph If we use the definition of the derivative above we can find the appropriate formula that defines the slope at any point on the graph 2 x h 5 x h 4 2 x 2 5 x 4 f x h f x f x lim lim h 0 h 0 h h 2 We can now simplify using basic algebra 2 x 2 2 2 xh 2h 2 5 x 5h 4 2 x 2 5 x 4 f x lim h 0 h Arizona State University Department of Mathematics and Statistics 2 Calculus Review Derivatives h 4 x 2h 5 4 xh 2h 2 5h f x lim lim lim 4 x 2h 5 4 x 5 h 0 h 0 h 0 h h Hence f x 4 x 5 is the formula that defines the derivative of the function f x 2 x 2 5 x 4 Let s apply the rule at the point 2 22 is on the graph 1 Make sure the point is on the graph Since f 2 2 2 5 2 4 8 10 4 22 the point is on the graph 2 2 Now evaluate the derivative At this point the derivative is f 2 4 2 5 8 5 13 We can say that the rate of change of y with respect to x at 2 22 is 13 f x h f x becomes tedious dangerously h 0 h fraught with calculation error and time consuming for anything beyond a quadratic We cannot even use it directly for most exponential and logarithmic functions We developed all the other formulas to save time and reduce using more complex algebra As much as I love to do it this limit process f x lim Examples Let s look at some many examples The only trick is to use the appropriate formulas 1 f x x 250 The definition would be tedious to use The power rule derivative more quickly It is 2 d n x n x n 1 allows us to find the dx d 250 x 250 x 249 dx f x 4 x5 2 x3 7 x 9 We can use the power rule on each of the 4 terms in this function Remember that f x 4 x 5 2 x 3 7 x 9 can be seen as the sum of four power functions So f x 4 5 x 4 2 3 x 2 7 1x 0 0 9 x 1 20 x 4 6 x 2 7 as we take the derivatives term byterm 3 f x 3 x 2 4 x 8 2 x3 5 This can be written as f x u v where u 3 x 2 4 x 8 and v 2 x 3 5 The formula that fits the function is the product rule d u v u v u v dx We can now find the derivative of each of these two new and simpler functions in the same way as before and then substituting them …